U.S. patent application number 09/740863 was filed with the patent office on 2001-09-27 for channel estimation by phase multiplexed complementary sequences.
This patent application is currently assigned to MITSUBISHI DENKI KABUSHIKI KAISHA. Invention is credited to Jechoux, Bruno, Rudolf, Marian.
Application Number | 20010024480 09/740863 |
Document ID | / |
Family ID | 9554149 |
Filed Date | 2001-09-27 |
United States Patent
Application |
20010024480 |
Kind Code |
A1 |
Rudolf, Marian ; et
al. |
September 27, 2001 |
Channel estimation by phase multiplexed complementary sequences
Abstract
The present invention relates to a method of estimating a
transmission or telecommunications channel in which method a
composite signal of complementary sequences such as:
.phi..sub.s,s(n)+.phi..sub.g,g(n)=k..delta.(n) (1) is used.
According to the present invention, said method is characterised in
that a pair of complementary sequences s(n) and g(n) is transmitted
after having multiplexed them in phase.
Inventors: |
Rudolf, Marian; (Rennes,
FR) ; Jechoux, Bruno; (Rennes, FR) |
Correspondence
Address: |
OBLON SPIVAK MCCLELLAND MAIER & NEUSTADT PC
FOURTH FLOOR
1755 JEFFERSON DAVIS HIGHWAY
ARLINGTON
VA
22202
US
|
Assignee: |
MITSUBISHI DENKI KABUSHIKI
KAISHA
TOKYO
JP
|
Family ID: |
9554149 |
Appl. No.: |
09/740863 |
Filed: |
December 21, 2000 |
Current U.S.
Class: |
375/343 |
Current CPC
Class: |
H04L 25/0226
20130101 |
Class at
Publication: |
375/343 |
International
Class: |
H03D 001/00; H04L
027/06 |
Foreign Application Data
Date |
Code |
Application Number |
Dec 30, 1999 |
FR |
9916849 |
Claims
1. Method of estimating a transmission or telecommunications
channel, in which method a composite signal of complementary
sequences such as: .phi..sub.s,s(n)+.phi..sub.g,g(n)=k..delta.(n)
(1) is used, the method being characterised in that a pair of
complementary sequences s(n) and g(n) is transmitted after having
multiplexed them in phase.
2. Method of estimating a transmission or telecommunications
channel according to claim 1, characterised in that the composite
signal is constructed from a pair of polyphase complementary
sequences s(n) and g(n) which are multiplexed in phase, this method
making it possible to exploit the property
.phi..sub.s,s(n)+.phi..sub.g,g(n) mentioned in the relation (1)
mentioned-in claim 1.
3. Method of estimating a transmission or telecommunications
channel according to claim 1, characterised in that the composite
signal is made up of two polyphase complementary sequences s(n) and
g(n), transmitted with a phase shift between them of 90.degree.,
i.e. the composite signal transmitted e(n) is in the form of the
relation (2) below: e(n)=e.sup.j.phi..(s(n)+j.g(n)) (2) with an
initial fixed and known phase shift .phi..
4. Device intended to generate the composite signal according to
relation (2) of claim 3, to implement the method of estimating a
transmission or telecommunications channel according to claim 1,
characterised in that it comprises a first generator (4) capable of
generating the first sequence s(n), with n varying from 0 to N-1,
the output of which is connected to the first input of an adder
(5), and a second generator (6) capable of generating the second
sequence g(n), with n varying from 0 to N-1, the output of which is
connected to the input of a first circuit (7) shifting phase by
90.degree., the output of which is connected to the second input of
the adder (5), the output of the adder (5) being connected to the
input of a second circuit (8), shifting phase by .phi. and which
delivers the composite signal e(n).
5. Estimation device which receives the received signal r(n)
corresponding to the composite signal according to claim 1,
characterised in that it comprises a processing circuit (9) made up
of two correlators (10 and 11) and an estimation device (12), the
input of the processing circuit (9) receives the signal r(n) and
applies the real part of it r.sub.1(n) to correlator (10), which
proceeds simultaneously to correlation with the sequences s(n) and
g(n), and applies the imaginary part r.sub.Q(n) to correlator (11),
which likewise proceeds to correlation with the two sequences s(n)
and g(n).
6. Estimation device according to claim 5, characterised in that
said estimation device (12) comprises four memories FiFo (13 to
16), memory (13) receiving the signal c.sub.1.sup.s(n), memory (14)
the signal c.sub.l.sup.g(n), memory (15) the signal
c.sub.Q.sup.s(n) and memory (16) the signal c.sub.Q.sup.g(n), the
outputs of memories (13 and 16) being respectively connected to the
two inputs of an adder circuit (17), whilst the outputs of memories
(14 and 15) are respectively connected to the two inputs of a
subtracter circuit (18), the output of circuit (17) delivering the
signal .omega..sub.1, whilst the output of circuit 18 delivers the
signal w.sub.2, these two signals being applied to a circuit (19)
for calculating .alpha. and .alpha..
Description
[0001] The present invention relates to a method of estimating a
transmission or telecommunications channel which uses complementary
sequences. The method results either in obtaining an optimal
estimation of the phase and of the attenuation in the case of a
single-path channel if the arrival time of the signal is known, or
in obtaining a very effective estimation of the delays, phases and
attenuations of the different paths in the case of a multipath
channel. The method also makes it possible to obtain an estimation
in the case of a channel of which it is not possible to distinguish
the different paths or in the case of a multipath channel, of which
one of the paths is very powerful in comparison with all the
others, as long as the arrival time of the signal is known.
[0002] In a telecommunications system, information circulates
between transmitters and receivers through-channels. In this
connection, FIG. 1 illustrates a model, which is discrete in time,
of the transmission chain between a transmitter 1 and a receiver 2
through a transmission channel 3. As a general rule, the
transmission channels can correspond to different physical, radio,
wire, optical media etc., and to different environments, fixed or
mobile communications, satellites, submarine cables, etc.
[0003] As a result of the multiple reflections of which the waves
emitted by transmitter 1 can be the object, channel 3 is a
multipath channel which is generally modelled as FIG. 1 indicates.
It is then considered to be a shift register 30 comprising L serial
cells (referred to by a subscript k able to take values of between
1 and L) and the contents of which are shifted towards the right of
FIG. 1 each time a symbol arrives at its input. The output of each
cell with the subscript k is applied to a filter 31 representing
the interference undergone by this output and introducing an
attenuation of the amplitude .alpha..sub.k, a phase shift
.alpha..sub.k and a delay r.sub.k. The outputs of the filters are
summed in a summer 32. The total impulse response thus obtained is
marked h(n).
[0004] The output of summer 32 is applied to the input of an adder
33 which receives, moreover, a random signal, modelled by a
Gaussian white noise, .omega.(n) which corresponds to the thermal
noise which is present in the telecommunications system. In FIG. 1,
the reference h(n) has been used, in channel 3, for the register
30, the filters 32 and the summer 33, followed by an adder which
adds the noise .omega.(n).
[0005] It will be understood that, if the transmitter 1 transmits
the signal e(n), the signal received r(n), in the receiver 2, is
thus: 1 r ( n ) = e ( n ) * h ( n ) + w ( n ) = e ( n ) * k = 1 L a
k ( n - r k ) j a k + w ( n ) = k = 1 L a k e ( n - r k ) j a k + w
( n )
[0006] In these expressions 2 h ( n ) = k = 1 L a k ( n - r k ) j
k
[0007] denotes the impulse response of the channel, .delta.(n)
being the Dirac impulse. The operator * denotes the convolution
product, defined by the following relation: 3 c ( n ) = a ( n ) * b
( n ) = m = - .infin. + .infin. a ( m ) b ( n - m )
[0008] Thus it is generally necessary to determine the
characteristics of channel 3, at a given moment, in order to thwart
the induced distortion of the transmitted signal e(n). In order to
obtain an estimation of h(n), i.e. of the coefficients
.alpha..sub.k, r.sub.k and .alpha..sub.k of the model of channel 3,
it is necessary to repeat this operation at a greater or lesser
frequency depending on the rate at which the characteristics of the
channel evolve.
[0009] A widespread method of estimating the channel consists in
transmitting, via transmitter 1, signals e(n) which are
predetermined and known-to receiver 2, and in comparing the signals
received r(n) in receiver 2, by means of a periodic or aperiodic
correlation, with those which are expected there in order to deduce
from them the characteristics of the channel. The aperiodic
correlation of two signals of length N has a total length 2N-1 and
is expressed, from the convolution product, by the relation: 4 a ,
b ( n ) a * ( - n ) * b ( n ) = m = 0 N - 1 a ( m ) ( b ( m + n ) )
( 1 ) , [ m ] = 0 , 1 , , N - 1
[0010] for two signals .alpha.(n) and h(n) of finite length N,
where the operator * denotes the complex conjugate operation.
[0011] The correlation of the received signal r(n) with the known
transmitted signal e(n) translates as:
r(n)*e.sup.*(-n)=[e(n)*h(n)+.omega.(n)]*e.sup.*(-n)
.phi..sub.e,r(n)=.phi..sub.e,e*h(n)+.phi..sub.e,w(n)
=.phi..sub.e,e(n)*h(n)+.phi..sub.e,w(n)
[0012] The result of the correlation operation constitutes the
estimation of the impulse response of the channel: the quality or
the precision of the estimation is aUl the better if
.phi..sub.e,r(n) tends towards h(n). The latter is directly
dependant on the choice of transmitted sequence e(n); to optimise
the estimation process, the signal e(n) should be chosen in such a
way that .phi..sub.e,e(n) tends towards k..delta.(n), k being a
real number, and that .phi..sub.e,.omega.(n)/.phi..sub.e,e(n) tends
towards O. In fact, in this case, the estimation of the channel
becomes:
.phi..sub.e,r(n)=k..delta.(n)*h(n)+.phi..sub.e,.omega.(n)
=k.h(n)+.phi..sub.e,.omega.(n)
.phi..sub.e,r(n).apprxeq.k.h(n)
[0013] It has been demonstrated that no single sequence exists for
which the function of aperiodic auto-correlation is equal to
.phi..sub.e,e(n), k,.delta.(n).
[0014] One object of the present invention consists in using pairs
of complementary sequences which have the property that the sum of
their auto-correlations is a perfect Dirac function. Let s(n) and
g(n), n=0,1, . . . , N-1 be a pair of complementary sequences:
.phi..sub.s,s(n)+.phi..sub.g,g(n)=k..delta.(n) (1)
[0015] Several methods of constructing such complementary sequences
are known in the literature: Golay complementary sequences,
polyphase complementary sequences, Welti sequences, etc. By way of
information, one will be able to refer, in this connection, to the
following technical documents which deal with the introduction to
complementary sequences and, in particular, to Golay complementary
sequences as well as to a Golay correlator:
[0016] 1) "On aperiodic and periodic complementary sequences" by
Feng K., Shiue P. J. -S., and Xiang Q., published in the technical
journal IEEE Transactions on Information Theory, Vol. 45, no. 1,
January 1999,
[0017] 2) "Korrelationssignale" by Luke H. -D, published in the
technical journal ISBN 3-540-545794, Springer-Verlag Heidelberg New
York, 1992,
[0018] 3) "Polypbase Complementary Codes" by R. L. Frank, published
in the technical journal IEEE Transactions on Information Theory,
November 1980, Vol. IT26, no. 6,
[0019] 4) "Multiphase Complementary Codes" by R. Sivaswamy,
published in the technical journal IEEE Transactions on Information
Theory, September 1978, Vol. IT-24, no. 5,
[0020] 5) "Efficient pulse compressor for Golay complementary
sequences" by S. Z. Budissin, published in the technical journal
Electronics Letters, Vol. 27, no. 3, January 1991,
[0021] 6) "Complementary Series" by M. J. Golay, published in the
technical journal IRE Trans; on Information Theory"Vol. IT-7, April
1961,
[0022] 7) "Efficient Golay Correlator" by B. M. Popovic, published
in the technical journal IEEE Electronics Letters, Vol. 35, no. 17,
August 1999.
[0023] Reference can also be made to the descriptions of the
documents U.S. Pat. Nos. 3,800,248, 4,743,753, 4,968,880,
5,729,612, 5,841,813, 5,862,182 and 5,961,463.
[0024] The property of complementary sequences in having a perfect
sum of autocorrelations is illustrated in FIG. 2, taking, by way of
example, a pair of Golay complementary sequences of length N=16
bits.
[0025] In FIG. 2 are plotted on the x-co-ordinates the time shifts
in relation to perfect synchronisation. The possible shifts are
numbered from 1 to 31 for the pair of sequences s(n) and g(n), and
on the y-co-ordinates the correlations from -5 to +35. The curve in
dashes corresponds to the auto-correlation .phi..sub.s,s(n) of the
sequence s(n); the curve in a dot-dash line to the auto-correlation
.phi..sub.g,g(n) of the sequence g(n): and the curve in an unbroken
line to the sum of the auto-correlations .phi..sub.s,s(n) and
.phi..sub.g,g(n). One can see that the curve in an unbroken line
merges with the axis of the x-co-ordinates between points 0 and 15
and points 17 and 31, but it corresponds practically to a Dirac
fimction between points 15 to 17.
[0026] The theoretically perfect auto-correlation properties of
these complementary sequences may, however, only be exploited if
their transmission can be ensured in such a manner that the
occurrence of inter-correlations .phi..sub.s,g(n) and /or
.phi..sub.g,s(n) is avoided.
[0027] According to one feature of the invention, a method is
provided of estimating a transmission or telecommunications
channel, in which method a composite signal of complementary
sequences is used and in which a pair of complementary sequences
s(n) and g(n) is transmitted after having multiplexed them in
phase.
[0028] According to another feature of the invention, a method is
provided of constructing the composite signal from a pair of
polyphase complementary sequences s(n) and g(n) which are
multiplexed in phase, this method making it possible to exploit the
property .phi..sub.s,s(n)+.phi..sub.g,g(n) mentioned in the
relation (1) above.
[0029] According to another feature, the composite signal is made
up of two polyphase complementary sequences s(n) and (g(n)
transmitted with a phase shift between them of 90.degree., i.e. the
transmitted composite signal e(n) is in the form of the relation
(2) below:
[0030] e(n)=e.sup.i.phi..(s(n)+j.g(n)) (2)
[0031] with an initial, fixed and known phase shift .phi..
[0032] In the case of binary complementary sequences s(n) and g(n),
with a number of phases P equal to 2, i.e. the case of Golay
complementary sequences, the transmitted signal e(n) is in the form
of a signal 2P-PSK, or 4-PSK, as FIG. 3 shows in the complex plane.
FIG. 3 represents, id the complex plane (R,I), the transmitted
composite signal e(n), of which the values 0 or 1 taken by each
component s(n), g(n) are respectively represented by the ends of a
corresponding segment S and G. Segments S and G are out of phase
with one another by II/2.
[0033] In the more general case of polyphase complementary
sequences with a number of phases P greater than 2, the transmitted
signal e(n) takes the form of a signal (2P)-PSK.
[0034] According to another feature, a device is provided which is
intended to generate the composite signal according to relation (2)
and which comprises a first generator capable of generating the
first sequence s(n), with n varying from 0 to N-1, the output of
which is connected to the first input of an adder, and a second
generator capable of generating the second sequence g(n), with n
varying from 0 to N-1, the output of which is connected to the
input of a first circuit shifting phase by 90.degree., the output
of which is connected to the second input of the adder, the output
of the adder being connected to the input of a second circuit
shifting phase by .phi. which delivers the composite signal.
[0035] The features of the present invention mentioned above, as
well as others, will appear more clearly in reading the description
of embodiments, said description being made in connection with the
attached drawings, amongst which:
[0036] FIG. 1 is a known diagram of a discrete model of a
transmission channel,
[0037] FIG. 2 is a known curve illustrating the auto-correlation of
two Golay complementary sequences and the sum of their
auto-correlations,
[0038] FIG. 3 illustrates a method of multiplexing in phase two
complementary sequences, according to the invention,
[0039] FIG. 4 is the diagram of an embodiment of the device
provided to generate the composite sequence of the invention,
[0040] FIG. 5 is a block diagram showing a circuit for processing
by correlation, connected in series with a device for estimating
the channel, the processing circuit receiving the signal r(n),
[0041] FIG. 6 is a block diagram showing an embodiment of a device
for single-path channel estimation, and
[0042] FIG. 7 is a block diagram showing another embodiment of a
device for multipath channel estimation.
[0043] The device shown in FIG. 4 is intended to produce the
composite signal according to relation (2), i.e.
e(n)=e.sup.j.phi..(s(n) j.g(n)) (2)
[0044] This device comprises a first generator 4 capable of
generating the first sequence s(n), with n varying from 0 to N-1,
the output of which is connected to the first input of an adder 5,
and a second generator 6 capable of generating the second sequence
g(n), with n varying from 0 to N-1, the output of which is
connected to the input of a first phase-shifting circuit, 7,
supplying a phase shift of 90.degree., the output of which is
connected to the second input of the adder 5, the output of the
adder 5 being connected to the input of a second phase-shifling
circuit 8 which supplies the phase shift .phi. and which delivers
the composite signal e(n).
[0045] FIG. 5 shows the general structure of a signal processing
circuit 9, to the input of which is applied the signal r(n)
received in the receiver 2, FIG. 1, coming from the transmission
channel 3.
[0046] Passing into a multipath channel, the total impulse response
of which is: 5 h ( n ) = k = 1 L a k ( n - r k ) j a k
[0047] the received signal r(n) becomes: 6 r ( n ) = k = 1 L a k j
a k e ( n - r k ) = k = 1 L a k j ( a k + ) ( s ( n - r k ) + j g (
n - r k ) ) = r I ( n ) + k r Q ( n )
[0048] The real and imaginary parts of the received signal r(n) are
expressed in the following manner: 7 r I ( n ) = Re { k = 1 L a k (
cos ( k + ) + j sin ( k + ) ) ( s ( n - r k ) + j g ( n - r k ) ) }
= k = 1 L ( a k cos ( k + ) s ( n - r k ) - a k sin ( k + ) g ( n -
r k ) ) r Q ( n ) = Im { k = 1 L a k ( cos ( k + ) + j sin ( k + )
) ( s ( n - r k ) + j g ( n - r k ) ) } = k = 1 L ( a k cos ( k + )
s ( n - r k ) + a k cos ( k + ) g ( n - r k ) ) ( 3 )
[0049] The processing circuit 9 is made up of two correlators 10
and 11 and an estimation device 12. The input of the processing
circuit 9 receives the signal r(n) and applies the real part
r.sub.1(n) to correlator 10 which proceeds separately to
correlation with the two sequences s(n) and g(n), and the imaginary
part r.sub.Q(n) to correlator 11 which proceeds likewise to
correlation with the two sequences s(n) and g(n),
[0050] Thus, at the respective outputs of correlators IO and 11,
signals are obtained which contain the contributions of the
auto-correlations of s(n) and g(n), and the contributions of their
inter-correlations, and which are mentioned below: 8 c I s ( n ) =
k = 1 L ( a k cos ( k + ) s , s ( n - r k ) - a k sin ( k + ) g , s
( n - r k ) ) 9 c I g ( n ) = k = 1 L ( a k cos ( k + ) s , g ( n -
r k ) - a k sin ( k + ) g , g ( n - r k ) ) c Q s ( n ) = k = 1 L (
a k sin ( k + ) s , g ( n - r k ) + a k cos ( k + ) g , s ( n - r k
) ) c Q g ( n ) = k = 1 L ( a k sin ( k + ) s , g ( n - r k ) + a k
cos ( k + ) g , g ( n - r k ) ) ( 4 )
[0051] of which the two first c.sub.1.sup.s(n) and
c.sub.1.sup.8(ii) are delivered by correlator 10, and the last two
c.sub.Q.sup.s(n) and c.sub.Q.sup.s(n) are delivered by correlator
11. These four signals are applied to the estimation device 12.
[0052] In a first case, that of device 12 of FIG. 6, it was
considered that the transmission channel 3 of FIG. 1 was a
single-path channel or even a multipath transmission channel, of
which it is not possible to distinguish the different paths, or a
multipath channel, of which one of the paths is very powerful in
comparison with all the other paths. In this case, the coefficient
L used in the relation of the preamble: 10 r ( n ) = e ( n ) * h (
n ) + w ( n ) = e ( n ) * k = 1 L a k ( n - r k ) j a k + w ( n ) =
k = 1 L a k e ( n - r k ) j a k + w ( n )
[0053] is equal to one, and if the arrival time r.sub.1 is known,
the correlation values obtained by the above relations (4) can be
combined in a simple manner, which makes it possible to determine
.alpha..sub.1 and .alpha..sub.1 via the estimation device 12 shown
in FIG. 6.
[0054] The estimation device of FIG. 6 comprises four memories FiFo
13 to 16, memory 13 receiving the signal c.sub.1.sup.s(n), memory
14 the signal c.sub.1.sup.g(n), memory 15 the signal
c.sub.Q.sup.s(n) and memory 16 the signal c.sub.Q.sup.g(n). For
each of these signals, all the 2N -1 correlation values centred on
the known arrival time of the signal r(n) are calculated and saved
in memory. The outputs of memories 13 and 16 are respectively
connected to the two inputs of an adder circuit 17, whilst the
outputs of memories 14 and 15 are respectively connected to the two
inputs of a subtracter circuit 18. The output of circuit 17
delivers the signal w.sub.1, whilst the output of circuit 18
delivers the signal .omega..sub.2. These two signals are applied to
a circuit 19 for calculating .alpha. and .alpha..
[0055] In calculating the signals .omega..sub.1 and .omega..sub.2
as the relations (5) indicate below:
.omega..sub.1(m)=(c.sub.1.sup.s(-m)).sup.*+c.sub.Q.sup.g(m)
=a.sub.1. cos
(.alpha..sub.1+.PHI.).((.phi..sub.s,s(-m)).sup.*+.phi..sub.g-
,g(m))+.alpha..sub.1. sin
(.alpha..sub.1+.phi.).(.phi..sub.s,g(m)-(.phi..s-
ub.g,s(-m)).sup.*)
.omega..sub.2=(m)=(c.sub.Q.sup.s(m)).sup.*-c.sub.1.sup.g(-m)
=.alpha..sub.1. sin
(.alpha..sub.1+.PHI.).(.phi..sub.s,s(m)+(.phi..sub.g,g-
(-m)).sup.*)+.alpha..sub.1. cos
(.alpha..sub.1+.phi.).(.phi..sub.g,s(m)-(.-
phi..sub.s,g(-m)).sup.*) (5)
[0056] where m=-N+1, -N+2, . . . N-2, N-1 is chosen as the index
for the correlation values calculated and saved in memory, in this
order.
[0057] With the two following relations:
.phi..sub.s,s(m)=.phi..sub.s,s.sup.*(-m)
.phi..sub.s,g(m)=.phi..sub.g,s.sup.*(-m)
[0058] which are valid for all s(n) and g(n) sequences, equation
(5) is simplified and one obtains:
.omega..sub.1(m)=.alpha..sub.1. cos
(.alpha..sub.1+.PHI.).(.phi..sub.s,s(m- )+.phi..sub.g,g(m))
.omega..sub.2(m)=.alpha..sub.1. sin
(.alpha..sub.1+.PHI.).(.phi..sub.s,s(m- )+.phi..sub.g,g(m)) (6)
[0059] These two signals are thus in the form of a Dirac weighted
by the channel coefficients, fvom which the attenuation and the
phase shift cain be obtained, in the calculating circuit 19, by the
relations: 11 1 = tan - 1 ( w 2 ( n - v ) w 1 ( n - v ) ) - a = w 1
( n - v ) cos ( 1 + ) or a = w 2 ( n - v ) sin ( 1 + )
[0060] with the initial known phase shift .phi..
[0061] In the more general case shown in FIG. 7, the signals
c.sub.1.sup.s(n), c.sub.l.sup.g(n), c.sub.Q.sup.s(n) and
c.sub.Q.sup.s(n) are applied respectively to the four inputs of a
circuit 20 which calculates the different coefficients
.alpha..sub.k and .alpha..sub.k, determines r.sub.k and delivers
them to its outputs.
[0062] Indeed, in the case of a multipath transmission channel, it
is not possible to eliminate the inter-correlation terms which one
had in the relations (4) above. With an appropriate circuit 20, it
is nevertheless possible to obtain estimations of coefficients of
the transmission channel.
[0063] Calculating from the equations (4), 12 w 1 ( n ) = c 1 s ( n
) + c Q g ( n ) = k = 1 L ( a k cos ( k + ) ( s , s ( n - r k ) + g
, g ( n - r k ) ) - a k sin ( k + ) ( s , g ( n - r k ) - g , s ( n
- r k ) ) ) w 2 ( n ) = c Q s ( n ) - c 1 g ( n ) = k = 1 L ( a k
sin ( k + ) ( s , s ( n - r k ) + g , g ( n - r k ) ) + a k sin ( k
+ ) ( g , s ( n - r k ) - s , g ( n - r k ) ) )
[0064] The two signals are in the formn of a Dirac weighted by the
channel coefficients plus other terms of inter-correlation between
the complementary sequences s(n) and g(n).
[0065] With 13 z 1 ( n ) = w 1 ( n ) + w 2 ( n ) = k = 1 L ( a k (
cos ( k + ) + sin ( k + ) ) ( s , s ( n - r k ) + g , g ( n - r k )
) + a k ( sin ( k + ) - cos ( k + ) ) ( g , s ( n - r k ) + s , g (
n - r k ) ) ) z 2 ( n ) = w 1 2 ( n ) + w 2 2 ( n ) = a k 2 ( s , s
( n - r k ) + g , g ( n - r k ) ) 2 + secondary terms
[0066] the delay r.sub.k are d erived in an obvious manner and the
attenuations and the phases can be determined by: 14 a k = z 2 ( n
) 2 N a k = - ( cos - 1 ( z 1 ( n ) 2 2 N ) + - .PI. 4 )
* * * * *